Sunday, 17 April 2016

The Logical Mr Carroll

Lewis Carroll, real name Charles
Dodgson, has several direct or indirect claims to fame or, some would say,
notoriety. He is best known now as the author of the ‘Alice’ books, written to
entertain Alice Liddell, the little daughter of Henry George Liddell, who was
the co-compiler of the still standard huge English dictionary of Ancient Greek (Henry, not Alice. (Duh)).
Carroll was a very fine photographer, and many of his photographs are of little
girls not wearing very much. In this post-Freudian witch-hunting age this
excites great suspicion among the prurient, though I don’t think anyone at the
time thought there was anything odd about it. Certainly Carroll himself would
have been horrified and disgusted at the suggestion there might have been
anything sexual (as of course there was) in his interest. We now know of course
— and the really suspect people are those who strenuously deny it — that all
human relations have a (perhaps unacknowledged) sexual element.

Anyway that’s more than enough
about that. Carroll / Dodgson lectured at Oxford University in Mathematics and
especially in logic. Some of his more elementary work in logic had much of the
wit of his stories and verses; here is an example:

(a) All babies are
illogical.

(b) Nobody is despised
who can manage a crocodile.

(c) Illogical persons
are despised.

As the subjects of this puzzle
are people, we take the universe as the set of all people. We will rewrite each
statement in the puzzle as an implication. First we define simpler statements,

B : it is a baby

L : it is logical

M : it can manage a crocodile

D : it is despised ,

where “it” in this context refers to a general
person. Then the three statements can be rephrased as

(a) B → ~L : If it is a baby
then it is not logical.

(b) M → ~D : If it can manage a
crocodile then it is not despised.

(c) ~L → D : If it is not
logical then it is despised.

Our aim is to use transitive reasoning several
times, stringing together a chain of implications using all the given
statements. We have an arrow pointing from B to ~L, and likewise an arrow
pointing from ~L to D; thus we are able to start with B and arrive at the
conclusion D. However, the second statement is still not utilized. But since
any implication is equivalent to its contrapositive, we may replace the second
statement with its contrapositive D → ~M. Then we get the transitive reasoning
chain

B → ~L →
D → ~M .

We reason that if B is true, then ~L is true, hence
D is true, and therefore ~M is true. Our ultimate conclusion is the statement

B → ~M :
If it is a baby then it cannot manage a crocodile .

In ordinary language we would more likely rephrase
this answer to the puzzle as

“No baby
can manage a crocodile.”

Alternatively, we could write the answer as the
contrapositive statement